2009-03 MTh SivaSiva Ganesan Rakash (2)

8/13/2019 2009-03 MTh SivaSiva Ganesan Rakash (2)

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Synchronization in MIMO-OFDMSystems

Master Thesis

by

SivaSiva Ganesan Rakash

Start date: 05. September 2008

End date: 06. March 2009Supervisor: Dipl.-Ing. Matthias Lieberei

Supervising Professor: Prof. Dr.-Ing. habil. Udo Zlzer
http://www.tuhh.de/http://www.tuhh.de/http://www.unibw-hamburg.de/


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Abstract

In this thesis, the time synchronization methods for MIMO-OFDM systemsare analyzed. In any communication system the transmitter and the receiverhave to be synchronized to each other in order to make efficient transmission.

Some special kind or format of sequences called pilot sequences are used forsynchronization. In case of multiple antenna systems, where different trans-mitters need to be synchronized to different receivers, the synchronizationis bit more complicated because of the multiple streams transmitted at thesame time in the same channel bandwidth. Some kind of orthogonal se-quences could be used, but the performance then directly depends on theorthogonality of such sequences. In multi-carrier systems like OFDM, thesynchronization is an important issue, because a small error in synchroniza-tion might lead to a significant loss in the system performance. Hence itis important to have a robust algorithm for synchronization. Besides, therobustness the amount of overhead introduced should also be considered.

Polyphase sequences have very good correlation property and a constantamplitude spectrum. In this thesis, polyphase sequences are used for syn-chronization. In case of MIMO-OFDM systems, cyclically shifted polyphasesequences are used to achieve synchronization. Effects of various systemparameters like noise level, channel order, oversampling etc. on the perfor-mance of the algorithm are analyzed. Also, the effect of the synchronizationalgorithm on the system performance is also evaluated. The mismatch be-tween the clocks of the DA and AD converters leads to a shift in the esti-mated channel position. By synchronization at regular intervals, these shiftsare identified and corrected automatically. Synchronization and channel es-timation are always closely related issues. The relation between these two

are analyzed and a method to use the channel estimation pilots to achievesynchronization over every OFDM symbol is proposed.

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Statement

Hereby I do state that this work has been prepared by myself and with the help which isreferred within this thesis.

Hamburg, March 06, 2009.

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Foreword

I take this opportunity to express my deepest gratitude to Dipl.-Ing. Matthias Lieberei for hisinvaluable support and the wholehearted confidence he has shown in me. My sincere thanksto Prof. Dr. Ing. Udo Zlzer for providing me this oppurtunity in his department. I thank all

those working in the ANT lab for their support and suggestions through out this thesis work.

Hamburg, March 06, 2009.

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CONTENTS ix

Contents

List of Figures xi

List of Tables xv

List of Symbols xvii

1 Introduction 1

2 System Theory 5

2.1 Linear Time Invariant systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Linearity Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.2 Time-invariance property . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Resolution of a Discrete-Time Signal into Impulses . . . . . . . . . . . . . . . . 6

2.3 Impulse response of a LTI system . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Physical Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.1 Cross correlation and autocorrelation . . . . . . . . . . . . . . . . . . . . 11

2.4.2 Normalization of Autocorrelation and Crosscorrelation sequences . . . . 12

2.4.3 Correlation of Complex Sequences . . . . . . . . . . . . . . . . . . . . . 12

2.5 Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5.2 Properties of the DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Introduction to OFDM 19

3.1 OFDM History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 OFDM Transmission Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 OFDM Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.1 Cyclic prefix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.2 Multi-path Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 Advantages of OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5 Disadvantages of OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.6 Introduction to MIMO Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.6.1 MIMO channel matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.6.2 Functions of MIMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Overview of MIMO-OFDM Laboratory System 314.1 Transmission chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1.1 Transmitter side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31


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LIST OF FIGURES xi

List of Figures

2.1 unit impulse function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

(a) (n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

(b) (n

k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Multiplication ofx(n) by impulse function . . . . . . . . . . . . . . . . . . . . . 7

(a) Random Signalx(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

(b) x(k) =x(n)(n k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Convolution as filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Convolution process - example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

(a) As filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

(b) As multipath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Plot of 1 Hz complex exponential function for one second . . . . . . . . . . . . . 152.6 1 Hz cosine wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.7 1 Hz cosine wave shifted by N/4 samples . . . . . . . . . . . . . . . . . . . . . . 16

3.1 Block Diagram for OFDM Transmission (baseband) . . . . . . . . . . . . . . . . 20

3.2 Overlapping subcarriers of OFDM spectrum . . . . . . . . . . . . . . . . . . . . 22

3.3 Spectrum of unit impulse function . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 Spectrum of channel impulse response . . . . . . . . . . . . . . . . . . . . . . . 24

(a) with two taps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

(b) with 8 taps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 Convolution with channel impulse response . . . . . . . . . . . . . . . . . . . . 25

3.6 MIMO Channel withNT transmit antennas and NR receive antennas . . . . . . 27

4.1 Block Diagram for laboratory MIMO-OFDM system . . . . . . . . . . . . . . . 32

5.1 Model for Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2 Polyphase sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

(a) Magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

(b) Differential Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.3 Autocorrelation of pphase sequence . . . . . . . . . . . . . . . . . . . . . . . . . 37


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xii LIST OF FIGURES

5.4 Autocorrelation of pphase sequence with cyclic repetition . . . . . . . . . . . . 38

5.5 Model to represent channel effect . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.6 Simplified model to represent channel effect . . . . . . . . . . . . . . . . . . . . 39

5.7 Autocorrelation of pphase sequence (with channel effect) . . . . . . . . . . . . . 39

5.8 Autocorrelation of pphase sequence (with frequency offset) . . . . . . . . . . . . 39

5.9 Timing Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.10 Timing Metric (no noise) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.11 Timing Metric (with noise) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.12 S m ooth in g filte r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.13 Smoothing of Timing Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.14 Timing Metric and ROI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.15 Matched filter output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.16 Matched filter output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.17 Moving FFT Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.18 Channel Impulse Response replicated at the sync position of MF output . . . . 48

5.19 Signal to Interference Ratio - No noise(Le = L + 1, Li= Nsp L 1) . . . . . . 495.20 Signal to Interference Ration - with noise(Le = L + 1, Li= synclength L 1) . 49

(a) SIR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

(b) zoomed version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.21 S IR (SNR = 0 dB) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.22 Matched filter output in 4 4 MIMO system . . . . . . . . . . . . . . . . . . . 515.23 Matched filter output in 4 4 MIMO system using cyclically shifted pphase

sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.24 Synchronization using orthogonal pphase sequences in a 4 4 MIMO system . 52(a) Coarse Synchronization output . . . . . . . . . . . . . . . . . . . . . . . . 52

(b) Matched Filtering output . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.25 Model to represent channel effect . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.26 Simplified model to represent channel effect . . . . . . . . . . . . . . . . . . . . 535.27 Channel Impulse Response replicated at the sync position of MF output . . . . 53

5.28 Channel Impulse Response replicated at the sync position of MF output (MIMO) 54

5.29 Sample pilot pattern for channel estimation in OFDM . . . . . . . . . . . . . . 54

5.30 Coarse Synchronization based on Cyclic Prefix . . . . . . . . . . . . . . . . . . 55

5.31 Fine Synchronization based on Channel Estimation . . . . . . . . . . . . . . . . 56

5.32 ADC clock faster than DAC clock . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.33 ADC clock slower than DAC clock . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.1 Synchronization using Cyclic Prefix . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.2 Synchronization based on cyclic prefix (no Noise, L = 0) . . . . . . . . . . . . . 60


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LIST OF FIGURES xiii

(a) Correlation output (entire signal) . . . . . . . . . . . . . . . . . . . . . . . 60

(b) Correlation output (Zoomed version) . . . . . . . . . . . . . . . . . . . . . 60

6.3 Synchronization based on cyclic prefix (SNR = 20 dB, L = 12) . . . . . . . . . 61

(a) With SNR = 10 d B (L=0) . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

(b) With L=12 (no noise) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.4 Synchronization based on cyclic prefix (L=0) . . . . . . . . . . . . . . . . . . . 61

(a) Synchronization probability across SNRs . . . . . . . . . . . . . . . . . . . 61

(b) Distribution of synchronization offset . . . . . . . . . . . . . . . . . . . . . 61

6.5 Synchronization based on cyclic prefix (no noise) . . . . . . . . . . . . . . . . . 62

(a) Synchronization probability across Ls . . . . . . . . . . . . . . . . . . . . 62


6.6 SNR gain from correlation over CP . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.7 Synchronization based on cyclic prefix (L=12) . . . . . . . . . . . . . . . . . . . 63

(a) Synchronization probability across SNRs . . . . . . . . . . . . . . . . . . . 63


6.8 Synchronization using CP (2L) . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.9 Mean of correlation metric over all OFDM symbols . . . . . . . . . . . . . . . . 64

6.10 Estimated position of the first sample of CP (SNR=6d B, = 1/50) . . . . . . 64

6.11 Estimated position of the first sample of CP (SNR=6d B, = 1/10) . . . . . . 65

6.12 Coarse Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.13 Coarse Synchronization output . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

(a) No noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

(b) 0 dB SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.14 Coarse Synchronization output (L=12, no noise - zoomed version) . . . . . . . . 66

6.15 Smoothing filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.16 Determination of alpha value . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.17 Determination of FD threshold (= 1, N = 70) . . . . . . . . . . . . . . . . . . 68

6.18 Determination of FD threshold (= 1/16, N = 70) . . . . . . . . . . . . . . . . 69

6.19 Determination of FD threshold (= 1, N= 140) . . . . . . . . . . . . . . . . . 69

6.20 MF output for negative exponential channel . . . . . . . . . . . . . . . . . . . . 70

(a) No noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

(b) 15 dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.21 MF output for random channel . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.22 Matched filter first tap distribution . . . . . . . . . . . . . . . . . . . . . . . . . 71

(a) 15 dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

(b) 0 dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.23 Threshold for MF for first tap (0.999) to be within the threshold . . . . . . . . 72


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xiv LIST OF FIGURES

6.24 Synchronization (to any of L+1 taps of the random channel) probability forvarious SNRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.25 Synchronization probability for various SNRs(negative exponential channel) . . 73

(a) Synchronize to any of the L+1 channel taps . . . . . . . . . . . . . . . . . 73

(b) Synchronize only to the first tap (kthr=0.5) . . . . . . . . . . . . . . . . . 73

6.26 Coarse synchronization output (ovs = 2) . . . . . . . . . . . . . . . . . . . . . . 73

6.27 Correlation of polyphase sequence (ovs = 1) . . . . . . . . . . . . . . . . . . . . 74

(a) L=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

(b) L=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.28 Correlation of polyphase sequence (ovs = 2) . . . . . . . . . . . . . . . . . . . . 74

(a) L=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

(b) L=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.29 Model for the effect of channel and oversampling on synchronization . . . . . . 74

6.30 Effect of oversampling and channel on Correlation output . . . . . . . . . . . . 76

6.31 Performance of synchronization algorithm in terms of BER (44MIMO system,= 5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.32 BER at different synchronization offsets (ideal channel estimation and equaliza-tion) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.33 Effect of synchronization offsets . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

(a) BER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

(b) Phase estimation error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.34 Ph as e Es tim ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

(a) Phase change across one OFDM symbol . . . . . . . . . . . . . . . . . . . 78

(b) zoomed version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.35 Effect of synchronization offsets on BER (for different number of estimation pilots) 79

6.36 Effect of synchronization offsets on BER (for different channel orders) . . . . . 79

6.37 Channel estimation based on MF output . . . . . . . . . . . . . . . . . . . . . . 81

6.38 BERs of synchronization using estimation pilots . . . . . . . . . . . . . . . . . . 81

6.39 MSDs of synchronization using estimation pilots . . . . . . . . . . . . . . . . . . 82

6.40 Position of the first tap of the CIR . . . . . . . . . . . . . . . . . . . . . . . . . 83

A.1 Smoothing filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

A.2 Coarse Synchronization output (L=12, no noise - zoomed version) . . . . . . . . 88


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LIST OF TABLES xv

List of Tables

2.1 Properties of the DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6.1 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59


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xvi LIST OF TABLES


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LIST OF SYMBOLS xvii

List of Symbols

a Arbitrary constant.c Complex constant.x Input to a system.

y Output of a system.w Noise. Unit impulse function.h System impulse response.rxx Autocorrelation ofx.rxy Cross correlation betweenx and y .Ex Energy of the signal x.xx Normalized autocorrelation.xy Normalized cross correlation. Phase.T OFDM symbol duration (without guard interval).

TG Guard interval.Ts OFDM symbol duration (with guard interval).f Separation between subcarriers.Nsc Number of subcarriers.Ncp Length of cyclic prefix.NT Number of transmitter antennas.NR Number of receiver antennas.L Channel order.max Maximum delay of the channel impulse response.BWcoh Coherent Bandwidth.d Additive white gaussian noise.

Sk Sent data symbol.Rk Received data symbol.Hk Channel coefficients (in frequency domain).

Hk Estimate of the channel coefficients.

Sk Estimate of the sent data symbol. oversampling factor.M Prime number used in the generation of polyphase sequence.Nsp Length of the synchronization sequence. Phase lag of 1 Hz frequency component of the polyphase sequence. Normalized timing metric.mean Mean of the normalized timing metric.

Threshold for the frame detection or coarse synchronization. Normalized cross correlation metric.


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xviii LIST OF SYMBOLS

kthr Threshold for matched filtering (to identify the first tap position). Coefficient of the IIR filter. 1 .f Fraction of the input that appear at the output of the IIR filter. Signal to Interference Ratio.n Synchronization offset.2 Equivalent noise of ICI and ISI due to synchronization offset.E{.} Expectation of the term within the braces.hLPF Impulse response of the low pass filter.


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1

Chapter 1

Introduction

The need for higher data rates has been increasing rapidly in the new generation data transmis-sion applications. The spectrum available is limited and in the multi-carrier system, OFDM,the spectrum is efficiently used. However, the maximum achievable data rate is limited to atight upper bound defined by the Shannon-Hartley theorem often referred to as the Shannonschannel capacity theorem. With the introduction of the multiple antenna transmission system,which uses spatial multiplexing, the limit is crossed and a very high data rate can be achieved.In OFDM systems and hence in MIMO-OFDM systems the synchronization is a key issue.A small synchronization error would lead to the loss of orthogonality of the subcarriers andtherefore the degradation of the system performance. Hence it is important to have a robustsynchronization algorithm and to know its effect on the system performance.

In our laboratory MIMO-OFDM system introduced in chapter 4, the transmitter and thereceiver antennas need to be time synchronized. The transmitter and the receiver side use thesame source for carrier frequency generation, hence the frequency is already synchronized.

In this thesis work, time synchronization is achieved using the circularly shifted polyphasesequences. The polyphase sequence has a very good autocorrelation property. The peak of theautocorrelation output of the polyphase sequence is used to achieve the time synchronization.The conventional synchronization is done in two steps,

1. Frame detection or coarse synchronization2. Matched filtering or fine synchronization

The polyphase sequence (let us say sync pilot and of size N) is transmitted twice at thebeginning of the signal. At the receiver, we make a running window correlation of every Nsamples with the next consecutive N samples. In the region of the polyphase sequence this willshow a strong increase in the correlation output, implying the presence of the sync pilots andhence the beginning of the first frame. In order to find the exact position of the first sampleof the sync pilot, we make a cross correlation of the known sync pilots with consecutive Nreceived samples in the region where the sync pilots are observed to be present. Exactly at the

start of the sync pilot we will observe a peak called sync peak and the samples before and afterit will be zeros. Knowing the position of the first sample of the sync pilot (second sync pilot),we can synchronize the data symbols. The thresholds for the coarse and fine synchronizations


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2 Chapter 1. Introduction

are optimized based on the maximum achievable synchronization probability at different SNRvalues.

In the case of MIMO-OFDM, polyphase sequences are transmitted from each of the transmitter

antenna but with a cyclic shift. Because of the good autocorrelation property, these cyclicallyshifted sequences would give a cross correlation output of zeros except at the multiples of thecyclic shift distance. As we already know the region where the sync pilot is present, we canchoose the correct peak from the cross correlation output.

When there is oversampling in the system, the correlation output results not only in a syncpeak surrounded by zeros but an oversampled version of the sync peak. That is we will have async peak at the position of the first sample of the sync pilot and other lesser amplitude samplessurrounding it due to the oversampling. Even though a small noise could change the positionof the sync peak, it is observed that we can synchronize to the position of the maximum inthis region and the BER of the resulting signal remains the same.

When there is a multi-path channel present, besides the single sync peak, we observe the delayedversions of the peak in the consecutive sample positions and its amplitude and phase are exactlyproportional to the channel impulse response. In case of the random channel it is difficult toidentify the first tap in the channel as the first tap may not be necessarily the strongest peak.The first tap could be too much attenuated such that it is difficult to differentiate it fromthe previous sample (zeros) positions especially at low SNR values. So we increase the CyclicPrefix (CP) to twice the channel order so that we have a tolerance limit to the synchronizationposition and could synchronize to any one of the channel taps. The CP is added before andafter every OFDM symbol to overcome small synchronization offset errors.

It is important to evaluate the performance of the synchronization algorithm in terms of BERthan the deviation from the correct sync position, because at the end we are interested in theBER of the demodulated signal. The effects of the synchronization errors on the received signalare studied mathematically and simulations are made to verify them in terms of BER.

The process of synchronization and channel estimation are related to each other. One processcould be achieved based on the outcome of the other. As the channel impulse response isobserved at the fine synchronization output, it could be used for channel estimation. Normallychannel estimation is done based on the pilots called Estimation Pilots(EP) which are insertedbetween the data symbols. The channel estimation based on synchronization is observed tobe better than the channel estimation based on EP. Instead of using some random symbols

in the position of the EP we could use the synchronization pilots in these positions and couldachieve synchronization using the overhead in the channel estimation process. A method touse the EP for synchronization has been proposed in this thesis. In this case, we could makesynchronization on every OFDM symbol without any extra overhead.

Due to the mismatch in the sampling clocks of the AD and DA converters, the estimated channeltends to shift either in the left or right depending upon the relative speed between the AD andDA converter clocks. This behavior is not desirable in situations where the channel is measuredover a long time to analyze its characteristics. The sync sequences are inserted between theOFDM data symbols and synchronization is made frequently (every OFDM symbol, in case ofsynchronization based on EP) so that this shift in the channel impulse response is identified

and corrected automatically.

In chapter 2, we give an introduction to the basic concepts of the system theory which are


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3

useful in interpretation of the methods and algorithms used in this thesis work. In chapter3, the OFDM and the MIMO-OFDM transmission techniques are described. Chapter 4 givesan overview of the structure of the laboratory MIMO-OFDM system and its specifications.In chapter 5 the time synchronization algorithm based on the cyclically shifted polyphasesequence is described. The model to describe the relation between the synchronization andchannel estimation is introduced. The methods to achieve synchronization during the channelestimation process and vice versa are explained. Also, the effect of the mismatch between theDAC and ADC clocks and the method to compensate the effect are discussed. In chapter 6 weevaluate the performance of the methods introduced in the chapter 5 and various parametersare optimized. Chapter 7 provides a brief summary of the work done and the results.


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4 Chapter 1. Introduction


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5

Chapter 2

System Theory

Every concept in system theory can be viewed differently depending upon the system envi-ronment, in which it is used, and also based on how it is processed in a given algorithm. Forexample if we take the process of convolution, it could be done for a continuous or a discretesignal, in time or frequency domain. The system impulse response, with which the input sig-nal is convoluted, can be used to represent an electronic circuit, a human vocal fold, a roomenvironment or the echos as in multi-path signal etc. This chapter introduces some of the con-cepts of system theory [1] in the view of Orthogonal Frequency Division Multiplexing (OFDM)communication system and synchronization of such OFDM systems.

2.1 Linear Time Invariant systems

Most of the systems in communication are assumed to be Linear Time Invariant (LTI) systems.These are the systems which satisfy the linearity property and the time invariance property.These properties will be defined and explained in this section.

2.1.1 Linearity Property

The general class of systems can be subdivided into linear systems and nonlinear systems. Asystem that satisfies the superposition principle is called linear system. Superposition princi-ple [1] requires that the response of the system to a weighted sum of the signals (inputs) beequal to the corresponding weighted sum of the responses (output) of the system to each ofthe individual input signals.

Letx1(n)andx2(n)be two arbitrary input sequences, anda1anda2be two arbitrary constants.Let T represent the system under consideration. Then T is linear if and only if,

T[a1x1(n) + a2x2(n)] =a1T[x1(n)] + a2T[x2(n)]. (2.1)

In the equation (2.1), ifa2 = 0, then,


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6 Chapter 2. System Theory

T[a1x1(n)] =a1T[x1(n)] (2.2)

equation (2.2) shows the multiplicative property of a linear system. That is, if the input to asystem is scaled by a factor a1, then the corresponding response is also scaled by a1.

In the equation (2.1), ifa1= a2= 1, then

T[x1(n) + x2(n)] =T[x1(n)] + T[x2(n)]. (2.3)

This equation (2.3) shows the additivity property of a linear system. That is, the response ofthe system to sum of the signals is equal to the corresponding sum of the response of the systemto each of the individual signals. The additivity and multiplicative properties constitute the

superposition principle as it applies to linear systems.

2.1.2 Time-invariance property

A system is called time-invariant or shift invariant, if its input-output characteristics do notchange with time. Lety(n)be the response of the systemTto the input signalx(n), representedas,

y(n) =T[x(n)] (2.4)

then if the same input signal is shifted by k units of time to yield,x(nk), and applied to thesystem (time invariant system), then the response of the system should also be delayed by kunits to yield y(n k). That is,

T[x(n k)] =y(n k) (2.5)

for the system to be time invariant, the equation (2.5) should hold for all possible values ofk .If it does not hold even for one value ofk, then the system is time variant.

2.2 Resolution of a Discrete-Time Signal into Impulses

The basic discrete-time signal is unit sample sequence. Any discrete-time signal can be rep-resented by the combinations of the unit sample sequences and its shifted versions. The unitsample sequence is denoted as (n)and is defined as,

(n) = 1 for n = 0

0 n = 0(2.6)

In general it could be written as,


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2.2 Resolution of a Discrete-Time Signal into Impulses 7

5 0 50

0.5

1

1.5

2

discrete time n

Amplitu

de

(a) (n)

5 0 50

0.5

1

1.5

2

discrete time n

Amplitu

de

k

(b) (n k)

Figure 2.1: unit impulse function

5 0 50

0.5

1

1.5

2

discrete time n

Amplitude

(a) Random Signalx

(n

)

5 0 50

0.5

1

1.5

2

discrete time n

Amplitude

k

(b) x

(k

) =x

(n

)

(n k

)

Figure 2.2: Multiplication ofx(n) by impulse function

(n k) =

1 for n = k

0 n =k (2.7)

These are shown in the Figures2.1(a)and 2.1(b).The term unit sample sequence refers to the

discrete version of the unit impulse function. In this thesis we consider only the discrete signalsand we use the terms unit sample sequence or unit impulse function or delta function to referto the discrete signal without ambiguity.

Suppose we have an arbitrary signal x(n) as shown in the Figure2.2(a),we multiply the twosequencesx(n)and (nk). Since(nk)is zero everywhere except atn = k (in our diagramk= 2), where delta function is unity, the result of our multiplication will have a value at n = kand zeros at all other positions. The value at n = k will be equal to x(k)as (n k)is one atn= k. Hence,

x(n)(n

k) =x(k)(n

k). (2.8)

Similarly, if we repeat this multiplication over all possible delays, < k


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x(n) =

k=

x(k)(n k). (2.9)

This equation2.9 represents the input sequence x(n)as a weighted sum of shifted unit samplesequences.

2.3 Impulse response of a LTI system

In a LTI system, if we know the response of the system to a unit impulse, then we couldcalculate the response of the system to arbitrary inputs. The output of the system is obtained

using convolution sum. Convolution is an operator, which operates between the arbitrary inputsignal and the system impulse response to give the corresponding output. Below we will seehow this convolution sum is arrived at.

Let, h(n,k) represent the impulse response of the LTI system under consideration. That is,it is the response y (n,k) of the system to the impulse sequence (n k) at n = k.

y(n,k) =h(n,k) =T[(n k)]. (2.10)

Let us take an arbitrary input sequence x(n), which could be decomposed as,

x(n) =

k=

x(k)(n k). (2.11)

If the input impulse to the LTI system is scaled by ck = x(k), then by multiplicative propertyof the LTI system, the response of the system is also multiplied by x(k). Hence,

x(k)h(n,k) =T[x(k)(n k)]. (2.12)

Therefore the response of the system tox(n)is,

y(n) = T[x(n)]

= T[

k=

x(k)(n k)]

=

k=x(k)T[(n k)]

=

k=

x(k)h(n,k) (2.13)


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Figure 2.3: Convolution as filtering

(a) As filter (b) As multipath

Figure 2.4: Convolution process - example

corresponds to the gain/loss in each path during its travel from the transmitter antenna to the

receiver antenna.

Convolution and Inter-sample interference:

If the number of sample in the system (channel) impulse response is L+1, then this impliesthat a sample transmitted at the time instant n, assuming zero propagation delay, will reachthe receiver antenna, not only at the time instant n but also at (n+1), (n+2), (n+3), .... (n+L),hence affecting the next consecutive L samples. Here, we have assumed that the time betweentwo consecutive samples to be equal to the delay between each path to the receiver. Here theconvolution of the input signal with the channel impulse response models the multi-path in-terference between the samples. This interference has to be equalized at the receiver. It will

be defined later in this chapter that the convolution in time domain is multiplication in thefrequency domain. Hence, a simple equalizer is to divide the spectrum of the received signal bythe channel transfer function fft(h(n)). We will see more about equalization in the chapter 3.


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2.4 Correlation 11

2.4 Correlation

A mathematical operation that closely resembles convolution is correlation. Like convolution,correlation is also an operator. This is used to measure the degree of similarity between twosignals. Very often we need to transmit two different sequences, say, x0(n) and x1(n), for0nN 1, each of length N. Mostly x0(n) is selected to be inverse ofx1(n). The signalreceived at the receiver may be represented as,

y(n) =xi(n) + w(n) i= 0,1 0 n N 1 (2.16)

w(n) represents the noise. The problem is to find whether x0(n) or x1(n) is present in thereceived signal y(n). The comparison is performed by the correlation of the received signal

with each of the transmitted sequences x0(n) and x1(n). If x0(n) was transmitted, then thecorrelation ofx0(n)with y(n)would result in a higher value than that ofx1(n).

2.4.1 Cross correlation and autocorrelation

Suppose we have two real sequences x(n) and y(n) each of which have finite energy. Thecorrelation ofx(n) and y (n)is a sequence rxy(l), which is defined as,

rxy(l) =

n=

x(n)y(n l) l= 0, 1,2, . . . (2.17)

The index l is the time shift parameter. The special case where y(n) = x(n) is called theautocorrelation.

rxx(l) =

n=

x(n)x(n

l) l= 0,

1,

2, . . . (2.18)

In dealing with finite duration sequences, we often use finite limits to express the cross corre-lation and autocorrelation. Ifx(n) and y (n)are of length N

rxy =

N|k|1n=i

x(n)y(n l) (2.19)

rxx=

N|k|1

n=i

x(n)x(n l) (2.20)

wherei = l,k = 0for l 0, and i = 0,k= l for l


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2.4.2 Normalization of Autocorrelation and Crosscorrelation sequences

The cross correlation sequence satisfies the condition that,

| rxy(l) |

rxx(0)ryy(0) =

ExEy. (2.21)

In the special case where y (n) =x(n),

| rxx(l) | rxx(0) =Ex. (2.22)

This implies that the autocorrelation sequence of a signal is maximum at its zero lag. At zero

lag(l= 0), the autocorrelation is the multiplication of the sequence with itself, which is equalto the sum of the squares (energy) of the signal. In case of cross correlation, the equation givesthe upper bound on the values of the cross correlation sequence. When the two sequences aresame, with one sequence affected by noise (say AWGN), we will get a maximum at zero lag,implying that a match has been found. This principle is used in the matched filter. We willdiscuss more about matched filtering in the chapter 5.

Even if any one or both of the signals involved in the correlation are scaled, the shape ofthe cross correlation does not change. Since scaling is unimportant when we try to find thesimilarity between the signals, we usually normalize the cross correlation and autocorrelationsequences to the range of -1 to +1. Hence, we divide them by the corresponding upper boundto normalize them. The normalized cross correlation is given by,

xy(l) = rxy(l)rxx(0)ryy(0)

(2.23)

by making x(n) =y(n), we get the normalized autocorrelation,

xx(l) = rxx(l)

rxx(0). (2.24)

2.4.3 Correlation of Complex Sequences

If one or both of the sequences being correlated are complex, then one among them is complexconjugated before the multiplication with the other. This means the phase of one of the signalis inversed and the magnitude remains same. So, when we multiply the sequences, the resultingproduct has phase equal to the phase difference between the two sequences and magnitudeequal to the product of the magnitude of the two sequences. This is illustrated below.

Ifx1(n)and x2(n)be two complex sequences. Then, the could be represented in the polar formas,

x1(n) =r1(n)ej1(n) (2.25)

x2(n) =r2(n)ej2(n) (2.26)


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2.5 Discrete Fourier Transform 13

The multiplication ofx1(n) and x2(n) is,

x1(n)x2(n) = r1(n)ej1(n)r2(n)ej2(n)

= r1(n)r2(n)ej(1(n)2(n). (2.27)

In the case, whenx1(n) =x2(n), we have,

x1(n)x2(n) = r1(n)e

j1(n)r1(n)ej1(n)

= r21(n). (2.28)

Hence, the cross correlation and autocorrelation for complex sequences is,

rxy =

N|k|1n=i

x(n)y(n l) (2.29)

rxx =

N|k|1n=i

x(n)x(n l) (2.30)

wherei = l,k = 0for l

0, and i = 0,k= l for l


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Table 2.1: Properties of the DFT

Property Time Domain Frequency Domain

Notation x(n),y(n) X(k),Y(k)

Periodicity x(n) =x(n + N) X(k) =X(k+ N)Linearity a1x1(n) + a2x2(n) a1X1(k) + a2X2(k)

Time reversal x(N n) X(N k)Circular time shift x((n l))N X(k)ej2kl/N

Circular frequency shift x(n)ej2ln/N X((k l))NComplex conjugate x(n) X(N k)

Circular convolution x1(n) x2(n) X1(k)X2(k)

Circular correlation x(n) y(n) X(k)Y(k)

Multiplication of two sequences x1(n)x2(n) 1

NX1(k)

X2(k)

Parsevals theorem N1

n=0 x(n)y(n) 1N

N1n=0 X(k)Y

(k)

2.5.1 Definition

The Discrete Fourier Transform of the sequence x(n) is given by,

X(k) =

N1n=0

x(n)ej2kn/N

k= 0,1,2, . . . , N 1 (2.32)and the Inverse DFT defined below transforms the frequency domain signal back to the timedomain.

x(n) =N1k=0

X(k)ej2kn/N n= 0,1,2, . . . , N 1 (2.33)

2.5.2 Properties of the DFT

The table2.1 [1] shows different properties of the DFT.

For the OFDM signal and for synchronization of the OFDM signal, the circular time shiftproperty is very important. We will discuss this elaborately.

Circular time shift property:

IfX(k)be the Npoint DFT ofx(n)represented as,

X(k) =N1

n=0

x(n)ej2kn/N k= 0,1,2, . . . , N

1 (2.34)

x(n)DFT X(k) (2.35)


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2.5 Discrete Fourier Transform 15

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

Figure 2.5: Plot of 1 Hz complex exponential function for one second

then,

x((n l))N DFT X(k)ej2kl/N (2.36)

To better look at this property, as defined earlier, Fourier transform decomposes the timedomain signal in to sum of weighted complex exponential function. That is, any given arbitrarysignal can be represented by a set of complex exponential function. These complex exponential

functions are uniquely defined by the frequency at which it is rotating. For simplicity, let usconsider the complex exponential function with a frequency1 Hz. Then this is a phase vectorrotating in anti-clockwise in the complex plane as shown in the Figure2.5.We assume the initialphase of this frequency component to be zero in the time domain signal under observation.

The magnitude of this phase vector shows the amplitude of this complex exponential function.The phase vector makes one complete circle (0 to 2) in one second, because the assumed fre-quency is1 Hz. This complex exponential function can be resolved into the real and imaginarypart and can be written as follow,

ej

= cos() +j sin() (2.37)

where, = 2(1)k/N, 0 k < N, 0


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0 100 200 3001

0.5

0

0.5

1

(degrees)

amplitude

Figure 2.6: 1 Hz cosine wave

0 100 200 300 4001

0.5

0

0.5

1

(degrees)

amplitude

Figure 2.7: 1 Hz cosine wave shifted by N/4 samples

f = 2lk/N (2.38)

In case of an OFDM symbol we always have a cyclic prefix and hence, when we are off by onesample during the demodulation, it would appear as a phase shift in the frequency domain andthe amplitude of the demodulated signal remains the same. We will discuss more about this inchapter 6.

2.6 Summary

In this chapter the basic concepts of signals and systems are introduced. Most of the systemsin communication are assumed to be linear time invariant. This assumption helps to model

the system using the impulse response of the system. Knowing the impulse response, we canfind the response of the system to any arbitrary input signal. This response to arbitrary inputsignal is obtained by convolution of the input signal with the impulse response of the system.


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2.6 Summary 17

Convolution could be viewed as a filtering operation or as a multi-path channel. The operatorsconvolution and correlation are closely related. Convolution without folding gives correlation.Correlation is a measure of degree of similarity between two sequences. The basic principlebehind the matched filter is, the correlation output maximizes at zero lag. Any signal couldbe resolved into weighted sum of complex exponential functions using Fourier transform. TheDFT transforms the discrete time domain signal into its frequency domain representation. Anycircular shift in time domain will appear as a corresponding phase shift in frequency domain.For more detailed explanation of these concepts, the reader is recommended to refer [1].


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19

Chapter 3

Introduction to OFDM

3.1 OFDM History

In a parallel data transmission technique, the available frequency band is divided into severalchannels. The signal for each of the channel is formed by modulating the data signal on separatecarriers (called sub-carriers in OFDM) of different frequency. As each channel occupies verynarrow bandwidth, amplitude and delay distortions can be easily equalized. In earlier multi-carrier system, the spectrum of each of the channels were kept non-overlapping in order toeliminate the inter-channel interference. In [2], Chang showed that a number of overlappingchannels each carrying a signaling rate of b, spaced b/2 apart in frequency could be transmittedwithout ICI and ISI, under certain conditions. When the number of channels are very large,then a transmission speed very close to Nyquist rate can be achieved. Later, in [3] Saltzberganalyzed the performance of such system and stated, The strategy of designing an efficientparallel system should concentrate more on reducing crosstalk between adjacent channels than

on perfecting the individual channels themselves, since the distortions due to the cross-talk tendto dominate. Also, for a large number of channels, the number of modulators and demodulatorsare increased proportionally. This makes the system very expensive. In [4], Salz et al proposed asystem which realized parallel data transmission using Discrete Fourier Transform. The multi-tone data signal is effectively, the Fourier Transform of the original time domain signal and thebank of coherent demodulators were replaced by the inverse Fourier Transform. Based on [4],Weinstein and Ebert in [5] described the use of DFT to perform the baseband modulation anddemodulation and investigated the effects of linear channel distortion. The use of the guardspace eliminated the most of the interference among the channels and between the adjacenttransmission blocks. Later, Peled and Ruiz in [6] introduced the cyclic prefix (CP). Theysuggested to fill the guard space with the cyclic extension of the OFDM symbol, which acts as

if it is performing cyclic convolution with the channel, provided the channel impulse responseis shorter than or equal to the length of the CP, thus preserving the convolution theorem andhence the orthogonality of the sub-carriers.


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20 Chapter 3. Introduction to OFDM

Figure 3.1: Block Diagram for OFDM Transmission (baseband)

3.2 OFDM Transmission Chain

The bits from the binary source are modulated using any one of the modulation techniqueslike BPSK, QPSK, QAM etc resulting in what we call as data symbols. The OFDM symbolis a weighted sum of windowed, orthogonal complex exponential functions. That is, we haveNsc sub-carriers (complex exponential functions) in frequency domain, separated by distanceoff. These frequency components (sub-carriers) are modulated by Nsc data symbols. TheIFFT of this modulated sub-carriers results in the corresponding time domain signal. The term

windowed refers to the fact that the complex exponential functions are observed only for theduration ofNscsamples, whereNscis the number of points in the fft window, here equal to thenumber of sub-carriers. Normally, not all the Nsc sub-carriers are modulated using the datasymbols, but some of the sub-carriers are modulated using the pilots symbols (symbols knownat the receiver) for the purpose of channel estimation. The number of pilots inserted betweendata depends on the allowable overhead and the required accuracy of the channel estimate.These pilots used for channel estimation are called Estimation Pilots (EP). Now we have anOFDM symbol of length Nsc. To this we add a cyclic prefix of length NG, which acts as aguard interval thus preventing the Inter Symbol Interference (ISI) and also preserving theorthogonality of the sub-carriers. After addition of the guard interval, the OFDM symbols isquadrature modulated and transmitted.

At the receiver, the received pass band signal is quadrature demodulated resulting in the OFDM


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3.3 OFDM Mathematical Model 21

symbol with guard interval. Then, the cyclic prefix in the guard interval, which has the ISIfrom the adjacent symbols is removed and given to the OFDM demodulator. The demodulatoris a FFT which would give the data and pilot symbols back in the frequency domain. Dueto the channel distortions, the symbols will not be received as it is transmitted. With thehelp of the pilot symbols, whose actual value is known at the receiver, the channel coefficientsin frequency domain can be estimated. Then the demodulated symbols are equalized. Theequalizer is a simple frequency domain equalizer, where we divide each of the received datasymbols by the channel coefficient of the corresponding sub-carrier. Then the data symbols aredemodulated based on the corresponding modulation technique (QPSK/QAM/...) used atthe transmiter to get back the actual bits.

3.3 OFDM Mathematical Model

Now we will describe the mathematical model [7] of the OFDM symbol and explain its physicalinterpretation. Let Sn,i represent the sequence of data symbols. n denote the OFDM symboltime index and i denote the sub-carrier frequency index, i = 1,2,..., Nsc. Let us consider anOFDM system using IFFT of size Nsc for modulation. The output of IFFT is a discrete timesignal with sampling interval T /Nsc. T is the duration of one OFDM symbol without guardinterval. Then the transmitted continuous complex baseband signal is given by

sn(t) =Nsc1

i=0

Sn,igi(t

nT)

gi(t) = e(j2ift)rect(

t T /2T

) (3.1)

whereSn,i is the data symbol transmitted in the nth OFDM symbol on the ith subcarrier. The

rect function in the equation (3.1) indicates that the complex exponential function is consideredonly in the region of 0 to T, that is from 0 . . . 2 (if the initial phase is zero and frequency1 Hz). As our sub-carriers (complex exponential functions) are restricted to a duration of T,the spectrum of the corresponding sub-carrier is extended infinitely. We know, multiplicationof the signal in time domain is convolution in the the frequency domain. The Fourier transform

of rect signal is a sinc pulse. Hence the sub-carriers in the frequency domain are convolutedwith the sinc signal. This is shown in the Figure 3.2

Gi(f) =Tsinc(T[f if]) (3.2)

From the Figure3.2 we see that the sinc pulse becomes zero at every 1/Tdistances from thecenter. Hence, if the sub-carriers are separated by a distance off= 1/T, then the inter-carrierinterference (ICI) can be avoided. That is the signalsgi(t) are orthogonal to each other.

Each of the data symbol are modulated on each individual complex exponential functions which

are present all along the duration 0 to Tin the time domain signal. Each of these exponentialfunctions are othogonal to each other. Hence, the duration of each of the data symbol is T. Ifwe want to transmit theseNsc symbols in a single-carrier system of same data rate, then the


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0.4

0.2

0

0.2

0.4

0.6

0.8

1

i+1 i+2ii1i2

Subcarrier spacing f

Figure 3.2: Overlapping subcarriers of OFDM spectrum

duration of each of the data symbol would be T /Nsc. Hence, in case of OFDM, the symbolduration is increased by a factor ofNsc. This increase in the duration of the symbol reduces

the inter symbol interference (ISI) compared to that of the single carrier system. Typically themaximum delay due to multi-path propagation is much smaller than the symbol duration T.

Samplingsn(t) in (3.1) with a time spacing T /Nsc gives,

sn,l = sn(nTs+ m T

Nsc)

=Nsc1

i=0Sn,ie

j2i 1Tl TNsc

sn,l =Nsc1i=0

Sn,iej2 il

Nsc . (3.3)

Equation3.3shows that the transmit signal could be obtained by the IDFT of the modulationsymbols.

3.3.1 Cyclic prefix

Due to the large symbol duration, the ISI is reduced but not completely zero. In order tocompletely remove ISI, a cyclic prefix is added to each of the OFDM symbol. The duration inwhich the cyclic prefix is added is called the guard interval (TG). To have ISI free transmission,


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3.3 OFDM Mathematical Model 23

the guard interval should be greater than the maximum multi-path delay (TG max). Hencethe total time duration of one OFDM symbol becomes,

Ts= T+ TG. (3.4)

The equation (3.1) then becomes,

sn(t) =Nsc1i=0

Sn,ig

i(t nTs)

g

i(t) = e(j2ift)rect

t Ts/2 + TG

Ts

. (3.5)

3.3.2 Multi-path Interference

In a multi-path environment, we not only receive the transmitted signal (amplitude and phasemodulated by channel) but also the delayed versions of the transmitted signal. This multi-pathchannel can be modeled by the channel impulse response. The received signal is the transmittedsignal convoluted with the channel impulse response and some noise added to it.

r(n) =s(n) h(n) + d(n) (3.6)

where, r(n) is the received signal, h(n) is the channel impulse response, d(n) is the AdditiveWhite Gaussian Noise (AWGN). For the time being let us consider a noiseless environment.

r(n) =s(n) h(n). (3.7)

If wes(n) =(n), thenr(n) =h(n). If the channel is not multi-path but only one direct path,thenh(n) =c, wherec is a complex constant, lets sayc = 1. Then the channel transfer functionlooks as shown in the Figure3.3. The Figure shows a constant magnitude spectrum. By theproperty of the Fourier transform, we know that convolution in time domain is multiplication

in frequency domain. That is the convolution of the input signal with the channel impulseresponse is equivalent to the multiplication of the channel transfer function (CTF) with thespectrum of the input signal. The CTF in our case infers that all the frequency componentsin the input signal are left unmodified or uniformly multiplied by a complex constant c (ifh(n) =c). So, the signal received at the receiver is received unaffected by the channel exceptfor a scaling.

But if the channel has a direct path and one delayed path, that is the channel is of order 1,then the channel impulse response is h(n) = [c1 c2]. Where c1 and c2 are complex constants.Ifc1 = 2 and c2 = 1, then the corresponding channel transfer function is shown in the Figure3.4(a).

Figure 3.4(a) shows that the spectrum is no more constant. This implies that the channelaffects different frequencies differently. That is the channel is frequency selective in nature.


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0.5 0 0.50

1

2

3

4

Normalized frequency

Magnitude

Figure 3.3: Spectrum of unit impulse function

0.5 0 0.50

1

2

3

4


Magnitude

(a) with two taps

0.5 0 0.50

2

4

6


Magnitude

(b) with 8 taps

Figure 3.4: Spectrum of channel impulse response

The frequency selectivity of the channel directly depends on the maximum multi-path delay(max) of the channel. The frequency selectivity increases with the increase in max. In otherterms, if we see the spectrum, the spectrum remains constant for a small frequency range calledCoherence Bandwidth. The coherence bandwidth is inversely proportional to the

max.

BWcoh= 1

max. (3.8)

We know that the bandwidth occupied by a symbol is inversely proportional to the symbolduration. That is f = 1T. Hence ifT max, then BWcoh f. Therefore, for each datasymbol(sub-carrier) the channel appears to be constant.

Now let us see how the cyclic prefix achieves ISI free transmission, how it preserves the orthog-onality and hence how the equalization process is simplified. As we described in the chapter 2,

the process of convolution can be seen as the addition of the signals from different paths.

Lets = [s1s2 s3 s4]be the OFDM symbol without guard interval.L = 2be the channel order,


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3.4 Advantages of OFDM 25

that is the channel impulse response has 3 taps, say h = [h0 h1 h2]. In order to completelyavoid ISI we need a cyclic prefix of at least 2 samples. After the addition of cyclic prefix,s = [ s3 s4 s1 s2 s3 s4]. For simplicity purpose, let us assume that before and after thisOFDM symbol, a sequence ofxs are transmitted. Hence, the sequence seen by the channel is[x . . . x x s3 s4 s1 s2 s3 s4 x x . . . x] . The convolution of this sequence with the channelimpulse response is shown in the Figure 3.5.

Figure 3.5: Convolution with channel impulse response

The red (left) and the green box constitutes the current OFDM symbol. The values in the regionof the green box show that, during this interval what we receive at the receiver is the circularconvolution of the OFDM symbol under consideration with the channel impulse response. Inthis region there is no ISI. Only in the region of the guard interval, marked by the red box, wehave ISI from the previous OFDM symbol, which will be thrown away by the receiver beforedemodulation. Also, in each row, which corresponds to each individual path, in the Figure3.5

we have a complete cycle of the signal but with a circular shift. If we say [s1 s2 s3 s4] hasonly one frequency component. Then in each path of the multi-path this frequency componentis received fully, though with a circular shift and a scaling by the channel. A circular shift intime domain corresponds to a phase change in the frequency domain. Similarly, all the otherfrequency components (if present) present in the transmitted signal are received in full cycleand remain orthogonal to each other at the receiver. Also, as each OFDM symbol transmittedis circularly convoluted with the channel impulse response, it is equivalent to multiplying thespectrum of the OFDM symbol by the channel transfer function. Hence, in order to equalizethe channel effect, we could simply divide the spectrum of the received OFDM symbol by thechannel transfer function. The channel transfer function can be estimated using the knownpilots symbols which are inserted between the data symbols at the transmitter, provided the

channel remains constant for one OFDM symbols duration.

3.4 Advantages of OFDM

1. The modulator and demodulators are efficiently realized using the IFFT and FFT re-spectively. Hence, separate filters for each sub-channel are not necessary.

2. Due to the overlapping sub-carriers, the bandwidth is used efficiently. A total symbolrate near Nyquist rate could be achieved.

3. ICI and ISI can be completely avoided.

4. Robust to multi-path fading (simple equalizer).


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3.5 Disadvantages of OFDM

1. More sensitive to frequency offset. Small frequency offset can cause loss of orthogonalitybetween the sub-carriers.

2. High peak to average power ratio. Sensitive to non-linear distortions.

3.6 Introduction to MIMO Systems

In Multiple Input Multiple Output (MIMO) technique, multiple antennas are used at thetransmitter and at the receiver. This method of multiple antenna transmission improves thespectrum efficiency or the reliability of the transmission without any additional bandwidth.Based on the requirements, MIMO techniques could be used for Spatial Multiplexing (SM) orfor achieving Diversity Gain (DG). It is also possible to make a trade off [8], between thesetwo, depending on the application.

3.6.1 MIMO channel matrix

At each of the receiver antennas, we receive the signals transmitted from each of the transmitterantennas. That means there is a channel (path) between each of the transmitter and each ofthe receiver antennas. If the transmitter and the receiver antennas are well spaced, then thereceived signals are sufficiently uncorrelated. This uncorrelation could also be achieved by usingantennas with different or orthogonal radiation pattern or by using antennas with differentpolarizations [9]. This means, each of the transmitter-receiver antenna pairs see different radiochannels. Let NT and NR be the number of transmit and receive antennas respectively. Theradio channel encountered between these antennas is the MIMO channel. If each of the channelis frequency non-selective, then a matrix of dimensionNRNT, with complex transfer factorsHi,j, can fully describe the channel behavior.

[H] =

H1,1 H1,2 H1,NTH2,1 H2,2 H2,NT

... ...

. . . ...

HNR,1 HNR,2 HNR,NT

If x represents the vector which consists of the streams transmitted from the transmitterantennas andy represents the received signal vector, then the transmit streams go through themulti-path channel and reach the receiver. This could be expressed as,

y= Hx + d (3.9)


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3.6 Introduction to MIMO Systems 27

Figure 3.6: MIMO Channel with NT transmit antennas andNR receive antennas

y1y2...

yNR

=

H1,1 H1,2 H1,NTH2,1 H2,2 H2,NT

... ...

. . . ...

HNR,1 HNR,2 HNR,NT

x1x2...

xNT

wherex is 1 NT vector and y is NR 1vector. d is the noise vector of same size as y . Fortime being if we consider no noise environment, then

y= Hx (3.10)

At the receiver if we want to decode the transmitted signal, then we need to solve

x= H1y (3.11)

The decoding of the transmitted signal depends on the channel matrix. Also, the MIMO ca-

pacity is a strong function of the singular value decomposition of the channel matrix [10]. Thenumber of different streams(Nn) that can be transmitted simultaneously depends on the rankof the channel matrix.

Nn= rank([H]) min(NT,NR) (3.12)

Let us consider the case where there is a strong line of sight transmission and no multi-pathpresent. Here the channel matrix is,

[H] =

1 1 11 1 11 1 1


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The system has rank one and the matrix is non-invertible. In this case we cannot transmitmore than 1 stream. This implies spatial multiplexing needs multi-path. Hence, for MIMO,multi-path is not enemy but ally [11]

3.6.2 Functions of MIMO

The MIMO functionalities can be divided into three main divisions [12]

1. Precoding:Precoding is a multilayer beamforming used in the MIMO systems. In con-ventional beamforming techniques, we transmit same signal stream from each of thetransmit antennas but with different weights and phase shifts, so that the signal poweris maximized at the receiver. In case of MIMO, where we have multiple antennas at thereceiver, we could not maximize the signal power at all the receivers using the conven-tional beamforming techniques. So precoding is done to maximize the power at each ofthe receiver antenna. Precoding needs the channel information.

2. Spatial Multiplexing:In spatial multiplexing, we transmit different streams of signalsfrom different transmit antennas. If we are able to decode all these streams indepen-dently at the receiver, then we have created NT parallel channels in the same frequencybandwidth and we gainNT 1 channels for free. This gain is called as the multiplexinggain. The maximum number of streams that could be transmitted depends on the rankof the channel matrix. Spatial Multiplexing could be done independent of the channelknowledge.

3. Diversity Coding: In diversity coding, the quality (reliability) of the antenna link isincreased by transmitting the same stream of signal from all the transmit antennas.For this purpose space time coding is used. The signal is redundantly transmitted fromdifferent transmit antennas. At each receiver antenna, the channel seen to each of thetransmit antennas are different. Hence some of the signals might survive the channel andcould be successfully decoded at the receiver. At the receiver, some combining techniques[13] like selection combining, maximum ratio combining, equal gain combining are alsoavailable. The former one is called transmit diversity and the later one is called receivediversity.

3.7 Summary

In this chapter a brief introduction of the OFDM transmission technique has been given.In OFDM the bandwidth is efficiently used using the overlapping orthogonal sub-carriers. Acyclic prefix (guard interval) of length equal to the channel order completely avoids the ISIand ICI. The modulation and the demodulation are effectively realized using the IFFT andFFT respectively. When the OFDM symbol duration is sufficiently larger than the maximumchannel tap delay, the bandwidth occupied by each of the sub-carrier is less than the coherencebandwidth. Hence for each sub-carrier the channel almost remains constant. As the channel isalmost constant for each sub-carrier, the equalization can be done by dividing the demodulated

data symbol by the corresponding estimated channel coefficient. Towards the end of the chapter,the MIMO technique has been introduced. The number of different streams that could betransmitted in parallel from different transmit antennas directly depends on the rank of the


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3.7 Summary 29

channel matrix. We also saw that the rank of the channel matrix not only depends on thenumber of transmit and receive antennas but also on the frequency selectivity (muli-pathnature) of the channel. Hence, in case of MIMO, the multi-path channel helps to improve theperformance of the system.


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32 Chapter 4. Overview of MIMO-OFDM Laboratory System

Figure 4.1: Block Diagram for laboratory MIMO-OFDM system

2. DA converter: The digital baseband I/Q signals are converted into analog basebandI/Q signals by the Digital to Analog (DA) converters. Each I and Q signals are sent tothe output as differential output signals. The DA converters used are of 14 bit resolutionwith a sampling rate of 125 Mss .

3. Local Oscillator (LO):The carrier frequency is generated using the LO. The generatedcarrier frequency is 2.412 GHz, which is the operating radio frequency of the laboratorysystem.

4. Quadrature Modulator:The analog baseband signal and the carrier frequency are fedto the quadrature modulator with a modulation bandwidth of 70 MHz, which modulatesthe I and Q signal on the carrier wave to produce the pass band radio signal.

5. Digital step attenuator: The signal output from the quadrature modulator is attenu-ated to a particular power level using digital step attenuator.

6. Amplifiers:The output signal from the attenuator is amplified using a power amplifierof gain 30 dB and a subsequent amplifier of gain 20 dB.

7. Transmit antennas: The signal thus created is transmitted as electromagnetic wavesusing standard magnet mount omni directional antenna. The signal is transmitted witha carrier frequency of 2.412 GHz and a bandwidth of 22 MHz.


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4.2 Need for synchronization 33

4.1.2 Receiver side

1. Receive Antenna: The receive antenna collects the energy from the electromagneticwaves emitted from the transmit antennas.

2. Channel filters: As the signal transmitted is known to be with a bandwidth of 22 MHzand at a carrier frequency of 2.412 GHz, the signals outside this band is filtered out usingthe channel filters.

3. Quadrature Demodulator: The filtered RF signal and the carrier frequency from thelocal oscillator (same LO used at the transmitter) are fed to the quadrature demodu-lator, which mixes the RF signal to the baseband. There is an inbuilt Automatic GainControl(AGC) circuit in it. Due to the initial transient time of the AGC, some part ofthe input signal will be damaged. The quadrature demodulator outputs the analog I andQ signals.

4. AD converter: The output analog baseband signals from the quadrature demodulatorare fed to the AD converter, which digitizes them. The AD converter is of 12 bit resolutionand of 100 Mss sampling rate.

5. Receiver PC: From the received digital baseband signal, the cyclic prefix in the guardinterval is removed and the data (and pilot) symbols are obtained using fft in MATLAB.From the received pilots signals, whose actual value is known at the receiver, the channelis estimated and the received signal is equalized. The data symbols are demodulatedusing BPSK or n-QAM as used in the transmitter side and the data bits are obtained.Comparing the received bits with the transmitted bits we could calculate the bit errorrate (BER).

4.2 Need for synchronization

The red lines in Figure 4.1 show the blocks between which the synchronization is necessary.If the AD and DA converters have different sampling clocks, then the mismatch in the clockspeed between these two would lead to either an additional sample or loss of sample dependingupon the relative speed between the two clocks. Also, the receiver side PC doesnt know whereexactly the first sample of the first OFDM symbol occurs. This is where the time synchroniza-tion comes into play. Normally besides the time synchronization, we need to make frequency

synchronization when there is a mismatch between the transmitter and receiver local oscilla-tors. But in our case, as we use the same local oscillator in the transmitter and the receiverside, the frequency offset is zero.


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34 Chapter 4. Overview of MIMO-OFDM Laboratory System


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35

Chapter 5

Time Synchronization Methods

In any transmission technique the signal transmitted from the transmitter antenna reachesthe receiver antenna after a propagation delay. Neither the instant of transmission nor thepropagation delay is known to the receiver. It might appear that at the receiver we could justcheck for the presence of the signal by its signal level (power) and find the beginning of thesignal. But this is not the case in reality. Before the reception of the actual transmitted signal,the channel noise is always received at the receiver. The Automatic Gain Control (AGC) circuitassumes this noise to be signal of lower power and amplifies it. Hence the presence of the signalcannot be determined based on its power level. Hence we need some kind of methodology tofind out the beginning of the signal. This is where the time synchronization comes into play.Besides this, there is also need to synchronize the frequency of the received signal with that

of the frequency of the transmitted signal, called frequency synchronization. Normally, thecarrier frequency generated by the local oscillator at the receiver is not exactly the same asthat of the carrier frequency generated by the local oscillator at the transmitter. This createsa frequency offset in the baseband signal. The frequency synchronization is very important forOFDM systems because a small frequency offset would cause loss of orthogonality between thesubcarriers and hence causing Inter Carrier Interference (ICI). But in our laboratory systemsame local oscillator is used at the transmitter and at the receiver end. So the frequency offset iszero. In this chapter (and in the remaining part of this thesis) we will discuss different methodsfor time synchronization.

5.1 Model for Synchronization

Our objective in time synchronization is to find the instant at which the first sample of thefirst OFDM symbol is arrived. One possible solution would be to add a synchronization pilot(sync pilot) in front of the signal to be transmitted. It is assumed that it is possible some howto identify the sync pilot at the receiver easily. Hence the signal after this would be the actualtransmitted signal and the signal before the sync pilot is the channel noise. For simulation

purpose, we normally transmit random signals of different lengths before the sync pilots, tomodel the channel noise amplified by the AGC before the reception of the actual signal. Thismodel is depicted in the Figure 5.1


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36 Chapter 5. Time Synchronization Methods

Figure 5.1: Model for Synchronization

20 40 600

0.5

1

1.5

2

time / frequency (time units / Hertz)

Magnitude(squareamplitude)

(a) Magnitude

0 20 40 604

2

0

2

4

time / frequency (time units / Hertz)

Differentialangle(radians)

(b) Differential Angle

Figure 5.2: Polyphase sequence

5.2 Introduction to polyphase sequences

From the model we could see that if the sync pilot is exactly identified then we could find theposition of the first sample of the first OFDM symbol and hence achieve synchronization. Now

the problem of synchronization is simplified to identification of the sync pilot at the receiver.For the time being we assume that the channel order (L) is 0. That is there is no multi-path.The effect of multi-path on synchronization will be discussed later in this chapter. Hence thesync pilot should be able to withstand the noise (AWGN) present in the channel. Sequenceswith good correlation property can be used for this purpose. We use polyphase sequences inthis project. The polyphase sequence of length Nsp is defined as,

s(k) =exp

j Mk

2

Nsp

(5.1)

where,k = 1 . . . N sp and Mis a relative prime factor ofN.

The Figure 5.2 shows the magnitude and relative phase between the consecutive samples ofthe polyphase sequence of length 64.

As it could be seen in the Figure 5.2, the x axis is labeled as time or frequency. If we considerthe polyphase (pphase) sequence to be a discrete time domain signal, then we have time. Ifwe consider the polyphase sequence to be a frequency domain signal (as we will use for oursynchronization method), then we have frequency. Only the phase of the samples changes butthe magnitude remains constant. From the equation (5.1) we see that the variablek is squared.This could be interpreted as one k for the time index and another for the frequency index. Inthis case at consecutive time instants, the consecutive frequency components are present but

with different phases. The phase difference between the consecutive samples is linear as shownin the Figure5.2(b). Here, the total phase difference change from the first sample to the lastsample is32. This is not a random value. It has to do with the relative prime factorMused


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5.2 Introduction to polyphase sequences 37

0 50 1000

10

20

30

40

50

60

70

delay (sample units)

Magnitude(squareamp

litude)

Figure 5.3: Autocorrelation of pphase sequence

in the generation of the polyphase sequence. For Nsp = 64, we took the largest prime factorof 64, which is 61. Hence the difference M Nsp =3. As explained before, the frequencycomponents can be interpreted as a phase vector rotating at the speed of its frequency. Inthe equation (5.1), if we leave the term MNsp , then at a given time instant (one k fixed), two

consecutive frequency components will differ exactly by one cycle (2). But when the frequencytermk is multiplied by the factor 6164 , then at the first time unit, the first frequency componentmakes 6164 part of one cycle. Hence it lags

=

1 61

64

(5.2)

in phase. Similarly the second frequency component will lag twice this amount in phase. Thephase lag of the second frequency component at the next time instant is four times the fractionof lag. Hence the first two samples of the polyphase sequence will differ by 3. This phasedifference between consecutive samples in the polyphase sequence increases linearly as the timeand the frequency increases along each sample. The difference in phase difference between theconsecutive samples will be twice the lag, one for frequency and one for time. Hence these

differences when accumulated for 64 samples would result in a difference of3 2.As the consecutive frequency components are of different frequencies and orthogonal to eachother, the dot product between all possible circular shifts of the sequence (except zero shift)would result in zero. If we see the autocorrelation(magnitude) of the sequence as shown in theFigure5.3 this is not the case. This is due to the fact that the autocorrelation of the sequenceis not an exact model for the dot products between the circularly shifted sequences. In orderto model this, let us assume that the polyphase sequence is repeated before and after theactual polyphase sequence under consideration. It will be shown later that this assumption isa reasonable one as we will any how have cyclic prefix in case of OFDM symbol structure andalso the sync pilots will be transmitted twice for the coarse synchronization purpose. In this

case we would observe a single peak (called sync peak) at the zero delay and zeros at all otherpositions as shown in the Figure5.4. Note that the x axis position of the peak is changed dueto the repetition of the polyphase sequence before and after the current polyphase sequence.


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230 240 250 260 270 2800

10

20

30

40

50

60

70


Magnitude(squareamp

litude)

Figure 5.4: Autocorrelation of pphase sequence with cyclic repetition

Figure 5.5: Model to represent channel effect

5.2.1 Effect of multi-path channel

If we have a multi-path channel, then the sequence is not only received once at the receiverbut also multiple delayed versions of the polyphase sequence are received at the receiver. Theeffect of the channel could be represented by the convolution of the channel impulse responsewith the transmitted pphase sequence. Then followed by autocorrelation at the receiver. Thisis shown in the Figure5.5.

Without loss of generality we could rearrange the blocks as shown in the Figure 5.6. Thisis equivalent to transmitting an impulse through the channel, which would give the channelimpulse response. Figure5.7 shows the effect of the channel on the autocorrelation property of

the polyphase sequence. When there is no multipath, the amplitude (and phase) of the syncpeak will be proportional to the channel coefficient (in the case of single path we have onlyone coefficient). In a multi-path environment the signal is received multiple times with differentdelays. These multiple reception correspond to each different path (multipath) and each path ischaracterized by a channel coefficient. Hence the correlation at the receiver would give multiplepeaks each proportional to the corresponding path. Hence, the output what we observe at thecorrelator output is directly proportional to the channel impulse response.

5.2.2 Effect of frequency offset

In OFDM systems, it is important to maintain the orthogonality between the subcarriers toavoid ICI. Though in our laboratory system we use the same local oscillators at the transmitter


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5.2 Introduction to polyphase sequences 39

Figure 5.6: Simplified model to represent channel effect

230 240 250 260 270 2800

10

20

30

40

50

60

70



Figure 5.7: Autocorrelation of pphase sequence (with channel effect)

230 240 250 260 270 2800

10

20

30

40

50

60

70



Figure 5.8: Autocorrelation of pphase sequence (with frequency offset)

and at the receiver and hence no frequency offset, we would describe the effect of frequencyoffset on the autocorrelation property of the pphase sequence just for completeness. Lets saywe have a frequency offset of 0.25 in our system. Then the polyphase sequence input to thecorrelator will look like,

s(k) =exp

j Mk

(k+0.25)N

. (5.3)


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As the frequency components are not at integer positions, the corresponding complex expo-nential functions will not make full cycle in the time domain window (Nsp) during which it isconsidered. Hence they are no more orthogonal and their dot product of the shifted sequenceswould not give zero. The correlator output is shown in the Figure5.8.

5.3 Synchronization algorithm

In general most of the synchronization algorithms achieve synchronization in two steps, namely,coarse synchronization and fine synchronization. In coarse synchronization the algorith

2009-03 MTh SivaSiva Ganesan Rakash (2)

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